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(* Title: HOL/Divides.thy 
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ID: $Id$ 

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Author: Lawrence C Paulson, Cambridge University Computer Laboratory 

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Copyright 1999 University of Cambridge 
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*) 
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header {* The division operators div, mod and the divides relation dvd *} 
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theory Divides 
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imports Nat Power Product_Type Wellfounded_Recursion 
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uses "~~/src/Provers/Arith/cancel_div_mod.ML" 
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begin 
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subsection {* Syntactic division operations *} 
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class div = times + 
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fixes div :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "div" 70) 
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fixes mod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "mod" 70) 

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begin 
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definition 
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dvd :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl "dvd" 50) 

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where 

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[code func del]: "m dvd n \<longleftrightarrow> (\<exists>k. n = m * k)" 

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end 

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subsection {* Abstract divisibility in commutative semirings. *} 

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class semiring_div = comm_semiring_1_cancel + div + 

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assumes mod_div_equality: "a div b * b + a mod b = a" 

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and div_by_0: "a div 0 = 0" 

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and mult_div: "b \<noteq> 0 \<Longrightarrow> a * b div b = a" 

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begin 

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text {* @{const div} and @{const mod} *} 
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lemma div_by_1: "a div 1 = a" 
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using mult_div [of 1 a] zero_neq_one by simp 
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lemma mod_by_1: "a mod 1 = 0" 

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proof  

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from mod_div_equality [of a one] div_by_1 have "a + a mod 1 = a" by simp 

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then have "a + a mod 1 = a + 0" by simp 

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then show ?thesis by (rule add_left_imp_eq) 

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qed 

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lemma mod_by_0: "a mod 0 = a" 

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using mod_div_equality [of a zero] by simp 

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lemma mult_mod: "a * b mod b = 0" 

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proof (cases "b = 0") 

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case True then show ?thesis by (simp add: mod_by_0) 

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next 

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case False with mult_div have abb: "a * b div b = a" . 

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from mod_div_equality have "a * b div b * b + a * b mod b = a * b" . 

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with abb have "a * b + a * b mod b = a * b + 0" by simp 

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then show ?thesis by (rule add_left_imp_eq) 

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qed 

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lemma mod_self: "a mod a = 0" 

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using mult_mod [of one] by simp 

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lemma div_self: "a \<noteq> 0 \<Longrightarrow> a div a = 1" 

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using mult_div [of _ one] by simp 

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lemma div_0: "0 div a = 0" 

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proof (cases "a = 0") 

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case True then show ?thesis by (simp add: div_by_0) 

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next 

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case False with mult_div have "0 * a div a = 0" . 

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then show ?thesis by simp 

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qed 

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lemma mod_0: "0 mod a = 0" 

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using mod_div_equality [of zero a] div_0 by simp 

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lemma mod_div_equality2: "b * (a div b) + a mod b = a" 
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unfolding mult_commute [of b] 

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by (rule mod_div_equality) 

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lemma div_mod_equality: "((a div b) * b + a mod b) + c = a + c" 

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by (simp add: mod_div_equality) 

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lemma div_mod_equality2: "(b * (a div b) + a mod b) + c = a + c" 

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by (simp add: mod_div_equality2) 

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text {* The @{const dvd} relation *} 
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lemma dvdI [intro?]: "a = b * c \<Longrightarrow> b dvd a" 
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unfolding dvd_def .. 

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lemma dvdE [elim?]: "b dvd a \<Longrightarrow> (\<And>c. a = b * c \<Longrightarrow> P) \<Longrightarrow> P" 

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unfolding dvd_def by blast 

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lemma dvd_def_mod [code func]: "a dvd b \<longleftrightarrow> b mod a = 0" 
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proof 

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assume "b mod a = 0" 

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with mod_div_equality [of b a] have "b div a * a = b" by simp 

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then have "b = a * (b div a)" unfolding mult_commute .. 

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then have "\<exists>c. b = a * c" .. 

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then show "a dvd b" unfolding dvd_def . 

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next 

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assume "a dvd b" 

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then have "\<exists>c. b = a * c" unfolding dvd_def . 

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then obtain c where "b = a * c" .. 

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then have "b mod a = a * c mod a" by simp 

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then have "b mod a = c * a mod a" by (simp add: mult_commute) 

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then show "b mod a = 0" by (simp add: mult_mod) 

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qed 

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lemma dvd_refl: "a dvd a" 

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unfolding dvd_def_mod mod_self .. 

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lemma dvd_trans: 

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assumes "a dvd b" and "b dvd c" 

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shows "a dvd c" 

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proof  

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from assms obtain v where "b = a * v" unfolding dvd_def by auto 

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moreover from assms obtain w where "c = b * w" unfolding dvd_def by auto 

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ultimately have "c = a * (v * w)" by (simp add: mult_assoc) 

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then show ?thesis unfolding dvd_def .. 

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qed 

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lemma zero_dvd_iff [noatp]: "0 dvd a \<longleftrightarrow> a = 0" 
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unfolding dvd_def by simp 
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lemma dvd_0: "a dvd 0" 

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unfolding dvd_def proof 

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show "0 = a * 0" by simp 

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qed 

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lemma one_dvd: "1 dvd a" 
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unfolding dvd_def by simp 

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lemma dvd_mult: "a dvd c \<Longrightarrow> a dvd (b * c)" 

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unfolding dvd_def by (blast intro: mult_left_commute) 

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lemma dvd_mult2: "a dvd b \<Longrightarrow> a dvd (b * c)" 

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apply (subst mult_commute) 

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apply (erule dvd_mult) 

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done 

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lemma dvd_triv_right: "a dvd b * a" 

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by (rule dvd_mult) (rule dvd_refl) 

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lemma dvd_triv_left: "a dvd a * b" 

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by (rule dvd_mult2) (rule dvd_refl) 

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lemma mult_dvd_mono: "a dvd c \<Longrightarrow> b dvd d \<Longrightarrow> a * b dvd c * d" 

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apply (unfold dvd_def, clarify) 

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apply (rule_tac x = "k * ka" in exI) 

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apply (simp add: mult_ac) 

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done 

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lemma dvd_mult_left: "a * b dvd c \<Longrightarrow> a dvd c" 

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by (simp add: dvd_def mult_assoc, blast) 

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lemma dvd_mult_right: "a * b dvd c \<Longrightarrow> b dvd c" 

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unfolding mult_ac [of a] by (rule dvd_mult_left) 

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end 
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subsection {* Division on @{typ nat} *} 
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text {* 
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We define @{const div} and @{const mod} on @{typ nat} by means 
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of a characteristic relation with two input arguments 
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@{term "m\<Colon>nat"}, @{term "n\<Colon>nat"} and two output arguments 
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@{term "q\<Colon>nat"}(uotient) and @{term "r\<Colon>nat"}(emainder). 
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*} 
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definition divmod_rel :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool" where 
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"divmod_rel m n q r \<longleftrightarrow> m = q * n + r \<and> (if n > 0 then 0 \<le> r \<and> r < n else q = 0)" 
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text {* @{const divmod_rel} is total: *} 
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lemma divmod_rel_ex: 
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obtains q r where "divmod_rel m n q r" 
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proof (cases "n = 0") 
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case True with that show thesis 
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by (auto simp add: divmod_rel_def) 
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next 
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case False 
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have "\<exists>q r. m = q * n + r \<and> r < n" 
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proof (induct m) 
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case 0 with `n \<noteq> 0` 
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have "(0\<Colon>nat) = 0 * n + 0 \<and> 0 < n" by simp 
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then show ?case by blast 
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next 
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case (Suc m) then obtain q' r' 
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where m: "m = q' * n + r'" and n: "r' < n" by auto 
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then show ?case proof (cases "Suc r' < n") 
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case True 
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from m n have "Suc m = q' * n + Suc r'" by simp 
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with True show ?thesis by blast 
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next 
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case False then have "n \<le> Suc r'" by auto 
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moreover from n have "Suc r' \<le> n" by auto 
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ultimately have "n = Suc r'" by auto 
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with m have "Suc m = Suc q' * n + 0" by simp 
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with `n \<noteq> 0` show ?thesis by blast 
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qed 
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qed 
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with that show thesis 
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using `n \<noteq> 0` by (auto simp add: divmod_rel_def) 
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qed 
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text {* @{const divmod_rel} is injective: *} 
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lemma divmod_rel_unique_div: 
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assumes "divmod_rel m n q r" 
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and "divmod_rel m n q' r'" 
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shows "q = q'" 
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proof (cases "n = 0") 
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case True with assms show ?thesis 
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by (simp add: divmod_rel_def) 
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next 
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case False 
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have aux: "\<And>q r q' r'. q' * n + r' = q * n + r \<Longrightarrow> r < n \<Longrightarrow> q' \<le> (q\<Colon>nat)" 
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apply (rule leI) 
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apply (subst less_iff_Suc_add) 
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apply (auto simp add: add_mult_distrib) 
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done 
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from `n \<noteq> 0` assms show ?thesis 
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by (auto simp add: divmod_rel_def 
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intro: order_antisym dest: aux sym) 
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qed 
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lemma divmod_rel_unique_mod: 
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assumes "divmod_rel m n q r" 
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and "divmod_rel m n q' r'" 
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shows "r = r'" 
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proof  
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from assms have "q = q'" by (rule divmod_rel_unique_div) 
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with assms show ?thesis by (simp add: divmod_rel_def) 
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qed 
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text {* 
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We instantiate divisibility on the natural numbers by 
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means of @{const divmod_rel}: 
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*} 
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instantiation nat :: semiring_div 

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begin 
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definition divmod :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where 
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[code func del]: "divmod m n = (THE (q, r). divmod_rel m n q r)" 
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definition div_nat where 
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"m div n = fst (divmod m n)" 
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definition mod_nat where 
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"m mod n = snd (divmod m n)" 
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lemma divmod_div_mod: 
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"divmod m n = (m div n, m mod n)" 
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unfolding div_nat_def mod_nat_def by simp 
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lemma divmod_eq: 
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assumes "divmod_rel m n q r" 
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263 
shows "divmod m n = (q, r)" 
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264 
using assms by (auto simp add: divmod_def 
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dest: divmod_rel_unique_div divmod_rel_unique_mod) 
25942  266 

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lemma div_eq: 
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268 
assumes "divmod_rel m n q r" 
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269 
shows "m div n = q" 
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using assms by (auto dest: divmod_eq simp add: div_nat_def) 
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271 

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lemma mod_eq: 
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273 
assumes "divmod_rel m n q r" 
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274 
shows "m mod n = r" 
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using assms by (auto dest: divmod_eq simp add: mod_nat_def) 
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lemma divmod_rel: "divmod_rel m n (m div n) (m mod n)" 
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proof  
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279 
from divmod_rel_ex 
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280 
obtain q r where rel: "divmod_rel m n q r" . 
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moreover with div_eq mod_eq have "m div n = q" and "m mod n = r" 
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282 
by simp_all 
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283 
ultimately show ?thesis by simp 
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284 
qed 
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285 

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286 
lemma divmod_zero: 
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287 
"divmod m 0 = (0, m)" 
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288 
proof  
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289 
from divmod_rel [of m 0] show ?thesis 
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290 
unfolding divmod_div_mod divmod_rel_def by simp 
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291 
qed 
25942  292 

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293 
lemma divmod_base: 
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294 
assumes "m < n" 
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295 
shows "divmod m n = (0, m)" 
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296 
proof  
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297 
from divmod_rel [of m n] show ?thesis 
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298 
unfolding divmod_div_mod divmod_rel_def 
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299 
using assms by (cases "m div n = 0") 
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(auto simp add: gr0_conv_Suc [of "m div n"]) 
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301 
qed 
25942  302 

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303 
lemma divmod_step: 
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304 
assumes "0 < n" and "n \<le> m" 
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305 
shows "divmod m n = (Suc ((m  n) div n), (m  n) mod n)" 
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306 
proof  
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307 
from divmod_rel have divmod_m_n: "divmod_rel m n (m div n) (m mod n)" . 
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308 
with assms have m_div_n: "m div n \<ge> 1" 
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by (cases "m div n") (auto simp add: divmod_rel_def) 
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from assms divmod_m_n have "divmod_rel (m  n) n (m div n  1) (m mod n)" 
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by (cases "m div n") (auto simp add: divmod_rel_def) 
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with divmod_eq have "divmod (m  n) n = (m div n  1, m mod n)" by simp 
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313 
moreover from divmod_div_mod have "divmod (m  n) n = ((m  n) div n, (m  n) mod n)" . 
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314 
ultimately have "m div n = Suc ((m  n) div n)" 
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315 
and "m mod n = (m  n) mod n" using m_div_n by simp_all 
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316 
then show ?thesis using divmod_div_mod by simp 
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317 
qed 
25942  318 

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text {* The ''recursionÂ´Â´ equations for @{const div} and @{const mod} *} 
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320 

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321 
lemma div_less [simp]: 
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322 
fixes m n :: nat 
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323 
assumes "m < n" 
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324 
shows "m div n = 0" 
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325 
using assms divmod_base divmod_div_mod by simp 
25942  326 

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327 
lemma le_div_geq: 
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328 
fixes m n :: nat 
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329 
assumes "0 < n" and "n \<le> m" 
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330 
shows "m div n = Suc ((m  n) div n)" 
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331 
using assms divmod_step divmod_div_mod by simp 
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332 

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333 
lemma mod_less [simp]: 
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334 
fixes m n :: nat 
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335 
assumes "m < n" 
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336 
shows "m mod n = m" 
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337 
using assms divmod_base divmod_div_mod by simp 
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338 

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339 
lemma le_mod_geq: 
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340 
fixes m n :: nat 
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341 
assumes "n \<le> m" 
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342 
shows "m mod n = (m  n) mod n" 
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343 
using assms divmod_step divmod_div_mod by (cases "n = 0") simp_all 
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344 

25942  345 
instance proof 
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346 
fix m n :: nat show "m div n * n + m mod n = m" 
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347 
using divmod_rel [of m n] by (simp add: divmod_rel_def) 
25942  348 
next 
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349 
fix n :: nat show "n div 0 = 0" 
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350 
using divmod_zero divmod_div_mod [of n 0] by simp 
25942  351 
next 
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352 
fix m n :: nat assume "n \<noteq> 0" then show "m * n div n = m" 
25942  353 
by (induct m) (simp_all add: le_div_geq) 
354 
qed 

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355 

25942  356 
end 
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357 

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358 
text {* Simproc for cancelling @{const div} and @{const mod} *} 
25942  359 

360 
lemmas mod_div_equality = semiring_div_class.times_div_mod_plus_zero_one.mod_div_equality [of "m\<Colon>nat" n, standard] 

26062  361 
lemmas mod_div_equality2 = mod_div_equality2 [of "n\<Colon>nat" m, standard] 
362 
lemmas div_mod_equality = div_mod_equality [of "m\<Colon>nat" n k, standard] 

363 
lemmas div_mod_equality2 = div_mod_equality2 [of "m\<Colon>nat" n k, standard] 

25942  364 

365 
ML {* 

366 
structure CancelDivModData = 

367 
struct 

368 

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val div_name = @{const_name div}; 
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val mod_name = @{const_name mod}; 
25942  371 
val mk_binop = HOLogic.mk_binop; 
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val mk_sum = ArithData.mk_sum; 
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val dest_sum = ArithData.dest_sum; 
25942  374 

375 
(*logic*) 

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25942  377 
val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}] 
378 

379 
val trans = trans 

380 

381 
val prove_eq_sums = 

382 
let val simps = @{thm add_0} :: @{thm add_0_right} :: @{thms add_ac} 

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383 
in ArithData.prove_conv all_tac (ArithData.simp_all_tac simps) end; 
25942  384 

385 
end; 

386 

387 
structure CancelDivMod = CancelDivModFun(CancelDivModData); 

388 

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389 
val cancel_div_mod_proc = Simplifier.simproc @{theory} 
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390 
"cancel_div_mod" ["(m::nat) + n"] (K CancelDivMod.proc); 
25942  391 

392 
Addsimprocs[cancel_div_mod_proc]; 

393 
*} 

394 

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395 
text {* code generator setup *} 
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396 

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397 
lemma divmod_if [code]: "divmod m n = (if n = 0 \<or> m < n then (0, m) else 
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398 
let (q, r) = divmod (m  n) n in (Suc q, r))" 
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399 
by (simp add: divmod_zero divmod_base divmod_step) 
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400 
(simp add: divmod_div_mod) 
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401 

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402 
code_modulename SML 
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403 
Divides Nat 
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404 

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405 
code_modulename OCaml 
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406 
Divides Nat 
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407 

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408 
code_modulename Haskell 
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409 
Divides Nat 
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410 

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411 

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412 
subsubsection {* Quotient *} 
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413 

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414 
lemmas DIVISION_BY_ZERO_DIV [simp] = div_by_0 [of "a\<Colon>nat", standard] 
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lemmas div_0 [simp] = semiring_div_class.div_0 [of "n\<Colon>nat", standard] 
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416 

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417 
lemma div_geq: "0 < n \<Longrightarrow> \<not> m < n \<Longrightarrow> m div n = Suc ((m  n) div n)" 
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418 
by (simp add: le_div_geq linorder_not_less) 
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419 

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420 
lemma div_if: "0 < n \<Longrightarrow> m div n = (if m < n then 0 else Suc ((m  n) div n))" 
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421 
by (simp add: div_geq) 
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422 

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423 
lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)" 
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424 
by (rule mult_div) simp 
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425 

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426 
lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)" 
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427 
by (simp add: mult_commute) 
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428 

25942  429 

430 
subsubsection {* Remainder *} 

431 

432 
lemmas DIVISION_BY_ZERO_MOD [simp] = mod_by_0 [of "a\<Colon>nat", standard] 

26100
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433 
lemmas mod_0 [simp] = semiring_div_class.mod_0 [of "n\<Colon>nat", standard] 
25942  434 

26100
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435 
lemma mod_less_divisor [simp]: 
fbc60cd02ae2
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436 
fixes m n :: nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
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437 
assumes "n > 0" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

438 
shows "m mod n < (n::nat)" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
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diff
changeset

439 
using assms divmod_rel unfolding divmod_rel_def by auto 
14267
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440 

26100
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using only an relation predicate to construct div and mod
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diff
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441 
lemma mod_less_eq_dividend [simp]: 
fbc60cd02ae2
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parents:
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442 
fixes m n :: nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
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parents:
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diff
changeset

443 
shows "m mod n \<le> m" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
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444 
proof (rule add_leD2) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
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445 
from mod_div_equality have "m div n * n + m mod n = m" . 
fbc60cd02ae2
using only an relation predicate to construct div and mod
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parents:
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446 
then show "m div n * n + m mod n \<le> m" by auto 
fbc60cd02ae2
using only an relation predicate to construct div and mod
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parents:
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changeset

447 
qed 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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448 

fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
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449 
lemma mod_geq: "\<not> m < (n\<Colon>nat) \<Longrightarrow> m mod n = (m  n) mod n" 
25942  450 
by (simp add: le_mod_geq linorder_not_less) 
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451 

26100
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using only an relation predicate to construct div and mod
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parents:
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diff
changeset

452 
lemma mod_if: "m mod (n\<Colon>nat) = (if m < n then m else (m  n) mod n)" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
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453 
by (simp add: le_mod_geq) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

454 

14267
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455 
lemma mod_1 [simp]: "m mod Suc 0 = 0" 
22718  456 
by (induct m) (simp_all add: mod_geq) 
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457 

25942  458 
lemmas mod_self [simp] = semiring_div_class.mod_self [of "n\<Colon>nat", standard] 
14267
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changeset

459 

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460 
lemma mod_add_self2 [simp]: "(m+n) mod n = m mod (n::nat)" 
22718  461 
apply (subgoal_tac "(n + m) mod n = (n+mn) mod n") 
462 
apply (simp add: add_commute) 

25942  463 
apply (subst le_mod_geq [symmetric], simp_all) 
22718  464 
done 
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465 

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466 
lemma mod_add_self1 [simp]: "(n+m) mod n = m mod (n::nat)" 
22718  467 
by (simp add: add_commute mod_add_self2) 
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468 

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469 
lemma mod_mult_self1 [simp]: "(m + k*n) mod n = m mod (n::nat)" 
22718  470 
by (induct k) (simp_all add: add_left_commute [of _ n]) 
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471 

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472 
lemma mod_mult_self2 [simp]: "(m + n*k) mod n = m mod (n::nat)" 
22718  473 
by (simp add: mult_commute mod_mult_self1) 
14267
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474 

26100
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using only an relation predicate to construct div and mod
haftmann
parents:
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475 
lemma mod_mult_distrib: "(m mod n) * (k\<Colon>nat) = (m * k) mod (n * k)" 
22718  476 
apply (cases "n = 0", simp) 
477 
apply (cases "k = 0", simp) 

478 
apply (induct m rule: nat_less_induct) 

479 
apply (subst mod_if, simp) 

480 
apply (simp add: mod_geq diff_mult_distrib) 

481 
done 

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482 

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483 
lemma mod_mult_distrib2: "(k::nat) * (m mod n) = (k*m) mod (k*n)" 
22718  484 
by (simp add: mult_commute [of k] mod_mult_distrib) 
14267
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diff
changeset

485 

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parents:
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486 
lemma mod_mult_self_is_0 [simp]: "(m*n) mod n = (0::nat)" 
22718  487 
apply (cases "n = 0", simp) 
488 
apply (induct m, simp) 

489 
apply (rename_tac k) 

490 
apply (cut_tac m = "k * n" and n = n in mod_add_self2) 

491 
apply (simp add: add_commute) 

492 
done 

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parents:
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diff
changeset

493 

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parents:
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494 
lemma mod_mult_self1_is_0 [simp]: "(n*m) mod n = (0::nat)" 
22718  495 
by (simp add: mult_commute mod_mult_self_is_0) 
14267
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parents:
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diff
changeset

496 

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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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parents:
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changeset

497 
(* a simple rearrangement of mod_div_equality: *) 
b963e9cee2a0
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498 
lemma mult_div_cancel: "(n::nat) * (m div n) = m  (m mod n)" 
22718  499 
by (cut_tac m = m and n = n in mod_div_equality2, arith) 
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500 

15439  501 
lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)" 
22718  502 
apply (drule mod_less_divisor [where m = m]) 
503 
apply simp 

504 
done 

14267
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changeset

505 

26100
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using only an relation predicate to construct div and mod
haftmann
parents:
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changeset

506 
subsubsection {* Quotient and Remainder *} 
14267
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507 

26100
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using only an relation predicate to construct div and mod
haftmann
parents:
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diff
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508 
lemma mod_div_decomp: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

509 
fixes n k :: nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

510 
obtains m q where "m = n div k" and "q = n mod k" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

511 
and "n = m * k + q" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

512 
proof  
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

513 
from mod_div_equality have "n = n div k * k + n mod k" by auto 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

514 
moreover have "n div k = n div k" .. 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

515 
moreover have "n mod k = n mod k" .. 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

516 
note that ultimately show thesis by blast 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

517 
qed 
14267
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parents:
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diff
changeset

518 

26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

519 
lemma divmod_rel_mult1_eq: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

520 
"[ divmod_rel b c q r; c > 0 ] 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

521 
==> divmod_rel (a*b) c (a*q + a*r div c) (a*r mod c)" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

522 
by (auto simp add: split_ifs mult_ac divmod_rel_def add_mult_distrib2) 
14267
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parents:
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diff
changeset

523 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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parents:
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diff
changeset

524 
lemma div_mult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::nat)" 
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

525 
apply (cases "c = 0", simp) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

526 
apply (blast intro: divmod_rel [THEN divmod_rel_mult1_eq, THEN div_eq]) 
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

527 
done 
14267
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
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diff
changeset

528 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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parents:
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diff
changeset

529 
lemma mod_mult1_eq: "(a*b) mod c = a*(b mod c) mod (c::nat)" 
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

530 
apply (cases "c = 0", simp) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

531 
apply (blast intro: divmod_rel [THEN divmod_rel_mult1_eq, THEN mod_eq]) 
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

532 
done 
14267
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
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diff
changeset

533 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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parents:
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changeset

534 
lemma mod_mult1_eq': "(a*b) mod (c::nat) = ((a mod c) * b) mod c" 
22718  535 
apply (rule trans) 
536 
apply (rule_tac s = "b*a mod c" in trans) 

537 
apply (rule_tac [2] mod_mult1_eq) 

538 
apply (simp_all add: mult_commute) 

539 
done 

14267
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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parents:
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changeset

540 

25162  541 
lemma mod_mult_distrib_mod: 
542 
"(a*b) mod (c::nat) = ((a mod c) * (b mod c)) mod c" 

543 
apply (rule mod_mult1_eq' [THEN trans]) 

544 
apply (rule mod_mult1_eq) 

545 
done 

14267
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

546 

26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

547 
lemma divmod_rel_add1_eq: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

548 
"[ divmod_rel a c aq ar; divmod_rel b c bq br; c > 0 ] 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

549 
==> divmod_rel (a + b) c (aq + bq + (ar+br) div c) ((ar + br) mod c)" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

550 
by (auto simp add: split_ifs mult_ac divmod_rel_def add_mult_distrib2) 
14267
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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diff
changeset

551 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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parents:
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diff
changeset

552 
(*NOT suitable for rewriting: the RHS has an instance of the LHS*) 
b963e9cee2a0
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paulson
parents:
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diff
changeset

553 
lemma div_add1_eq: 
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

554 
"(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)" 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

555 
apply (cases "c = 0", simp) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

556 
apply (blast intro: divmod_rel_add1_eq [THEN div_eq] divmod_rel) 
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

557 
done 
14267
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

558 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
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diff
changeset

559 
lemma mod_add1_eq: "(a+b) mod (c::nat) = (a mod c + b mod c) mod c" 
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

560 
apply (cases "c = 0", simp) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

561 
apply (blast intro: divmod_rel_add1_eq [THEN mod_eq] divmod_rel) 
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

562 
done 
14267
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

563 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
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diff
changeset

564 
lemma mod_lemma: "[ (0::nat) < c; r < b ] ==> b * (q mod c) + r < b * c" 
22718  565 
apply (cut_tac m = q and n = c in mod_less_divisor) 
566 
apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto) 

567 
apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst) 

568 
apply (simp add: add_mult_distrib2) 

569 
done 

10559
d3fd54fc659b
many new div and mod properties (borrowed from Integ/IntDiv)
paulson
parents:
10214
diff
changeset

570 

26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

571 
lemma divmod_rel_mult2_eq: "[ divmod_rel a b q r; 0 < b; 0 < c ] 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

572 
==> divmod_rel a (b*c) (q div c) (b*(q mod c) + r)" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

573 
by (auto simp add: mult_ac divmod_rel_def add_mult_distrib2 [symmetric] mod_lemma) 
14267
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

574 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
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diff
changeset

575 
lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)" 
22718  576 
apply (cases "b = 0", simp) 
577 
apply (cases "c = 0", simp) 

26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

578 
apply (force simp add: divmod_rel [THEN divmod_rel_mult2_eq, THEN div_eq]) 
22718  579 
done 
14267
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

580 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

581 
lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)" 
22718  582 
apply (cases "b = 0", simp) 
583 
apply (cases "c = 0", simp) 

26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

584 
apply (auto simp add: mult_commute divmod_rel [THEN divmod_rel_mult2_eq, THEN mod_eq]) 
22718  585 
done 
14267
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

586 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

587 

25942  588 
subsubsection{*Cancellation of Common Factors in Division*} 
14267
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

589 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

590 
lemma div_mult_mult_lemma: 
22718  591 
"[ (0::nat) < b; 0 < c ] ==> (c*a) div (c*b) = a div b" 
592 
by (auto simp add: div_mult2_eq) 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

593 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

594 
lemma div_mult_mult1 [simp]: "(0::nat) < c ==> (c*a) div (c*b) = a div b" 
22718  595 
apply (cases "b = 0") 
596 
apply (auto simp add: linorder_neq_iff [of b] div_mult_mult_lemma) 

597 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

598 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

599 
lemma div_mult_mult2 [simp]: "(0::nat) < c ==> (a*c) div (b*c) = a div b" 
22718  600 
apply (drule div_mult_mult1) 
601 
apply (auto simp add: mult_commute) 

602 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

603 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

604 

25942  605 
subsubsection{*Further Facts about Quotient and Remainder*} 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

606 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

607 
lemma div_1 [simp]: "m div Suc 0 = m" 
22718  608 
by (induct m) (simp_all add: div_geq) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

609 

25942  610 
lemmas div_self [simp] = semiring_div_class.div_self [of "n\<Colon>nat", standard] 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

611 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

612 
lemma div_add_self2: "0<n ==> (m+n) div n = Suc (m div n)" 
22718  613 
apply (subgoal_tac "(n + m) div n = Suc ((n+mn) div n) ") 
614 
apply (simp add: add_commute) 

615 
apply (subst div_geq [symmetric], simp_all) 

616 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

617 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

618 
lemma div_add_self1: "0<n ==> (n+m) div n = Suc (m div n)" 
22718  619 
by (simp add: add_commute div_add_self2) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

620 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

621 
lemma div_mult_self1 [simp]: "!!n::nat. 0<n ==> (m + k*n) div n = k + m div n" 
22718  622 
apply (subst div_add1_eq) 
623 
apply (subst div_mult1_eq, simp) 

624 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

625 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

626 
lemma div_mult_self2 [simp]: "0<n ==> (m + n*k) div n = k + m div (n::nat)" 
22718  627 
by (simp add: mult_commute div_mult_self1) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

628 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

629 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

630 
(* Monotonicity of div in first argument *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

631 
lemma div_le_mono [rule_format (no_asm)]: 
22718  632 
"\<forall>m::nat. m \<le> n > (m div k) \<le> (n div k)" 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

633 
apply (case_tac "k=0", simp) 
15251  634 
apply (induct "n" rule: nat_less_induct, clarify) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

635 
apply (case_tac "n<k") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

636 
(* 1 case n<k *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

637 
apply simp 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

638 
(* 2 case n >= k *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

639 
apply (case_tac "m<k") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

640 
(* 2.1 case m<k *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

641 
apply simp 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

642 
(* 2.2 case m>=k *) 
15439  643 
apply (simp add: div_geq diff_le_mono) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

644 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

645 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

646 
(* Antimonotonicity of div in second argument *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

647 
lemma div_le_mono2: "!!m::nat. [ 0<m; m\<le>n ] ==> (k div n) \<le> (k div m)" 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

648 
apply (subgoal_tac "0<n") 
22718  649 
prefer 2 apply simp 
15251  650 
apply (induct_tac k rule: nat_less_induct) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

651 
apply (rename_tac "k") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

652 
apply (case_tac "k<n", simp) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

653 
apply (subgoal_tac "~ (k<m) ") 
22718  654 
prefer 2 apply simp 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

655 
apply (simp add: div_geq) 
15251  656 
apply (subgoal_tac "(kn) div n \<le> (km) div n") 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

657 
prefer 2 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

658 
apply (blast intro: div_le_mono diff_le_mono2) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

659 
apply (rule le_trans, simp) 
15439  660 
apply (simp) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

661 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

662 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

663 
lemma div_le_dividend [simp]: "m div n \<le> (m::nat)" 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

664 
apply (case_tac "n=0", simp) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

665 
apply (subgoal_tac "m div n \<le> m div 1", simp) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

666 
apply (rule div_le_mono2) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

667 
apply (simp_all (no_asm_simp)) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

668 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

669 

22718  670 
(* Similar for "less than" *) 
17085  671 
lemma div_less_dividend [rule_format]: 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

672 
"!!n::nat. 1<n ==> 0 < m > m div n < m" 
15251  673 
apply (induct_tac m rule: nat_less_induct) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

674 
apply (rename_tac "m") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

675 
apply (case_tac "m<n", simp) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

676 
apply (subgoal_tac "0<n") 
22718  677 
prefer 2 apply simp 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

678 
apply (simp add: div_geq) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

679 
apply (case_tac "n<m") 
15251  680 
apply (subgoal_tac "(mn) div n < (mn) ") 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

681 
apply (rule impI less_trans_Suc)+ 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

682 
apply assumption 
15439  683 
apply (simp_all) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

684 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

685 

17085  686 
declare div_less_dividend [simp] 
687 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

688 
text{*A fact for the mutilated chess board*} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

689 
lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))" 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

690 
apply (case_tac "n=0", simp) 
15251  691 
apply (induct "m" rule: nat_less_induct) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

692 
apply (case_tac "Suc (na) <n") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

693 
(* case Suc(na) < n *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

694 
apply (frule lessI [THEN less_trans], simp add: less_not_refl3) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

695 
(* case n \<le> Suc(na) *) 
16796  696 
apply (simp add: linorder_not_less le_Suc_eq mod_geq) 
15439  697 
apply (auto simp add: Suc_diff_le le_mod_geq) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

698 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

699 

14437  700 
lemma nat_mod_div_trivial [simp]: "m mod n div n = (0 :: nat)" 
22718  701 
by (cases "n = 0") auto 
14437  702 

703 
lemma nat_mod_mod_trivial [simp]: "m mod n mod n = (m mod n :: nat)" 

22718  704 
by (cases "n = 0") auto 
14437  705 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

706 

26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

707 
(* Antimonotonicity of div in second argument *) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

708 
lemma div_le_mono2: "!!m::nat. [ 0<m; m\<le>n ] ==> (k div n) \<le> (k div m)" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

709 
apply (subgoal_tac "0<n") 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

710 
prefer 2 apply simp 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

711 
apply (induct_tac k rule: nat_less_induct) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

712 
apply (rename_tac "k") 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

713 
apply (case_tac "k<n", simp) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

714 
apply (subgoal_tac "~ (k<m) ") 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

715 
prefer 2 apply simp 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

716 
apply (simp add: div_geq) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

717 
apply (subgoal_tac "(kn) div n \<le> (km) div n") 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

718 
prefer 2 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

719 
apply (blast intro: div_le_mono diff_le_mono2) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

720 
apply (rule le_trans, simp) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

721 
apply (simp) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

722 
done 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

723 

fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

724 
lemma div_le_dividend [simp]: "m div n \<le> (m::nat)" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

725 
apply (case_tac "n=0", simp) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

726 
apply (subgoal_tac "m div n \<le> m div 1", simp) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

727 
apply (rule div_le_mono2) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

728 
apply (simp_all (no_asm_simp)) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

729 
done 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

730 

fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

731 
(* Similar for "less than" *) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

732 
lemma div_less_dividend [rule_format]: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

733 
"!!n::nat. 1<n ==> 0 < m > m div n < m" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

734 
apply (induct_tac m rule: nat_less_induct) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

735 
apply (rename_tac "m") 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

736 
apply (case_tac "m<n", simp) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

737 
apply (subgoal_tac "0<n") 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

738 
prefer 2 apply simp 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

739 
apply (simp add: div_geq) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

740 
apply (case_tac "n<m") 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

741 
apply (subgoal_tac "(mn) div n < (mn) ") 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

742 
apply (rule impI less_trans_Suc)+ 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

743 
apply assumption 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

744 
apply (simp_all) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

745 
done 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

746 

fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

747 
declare div_less_dividend [simp] 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

748 

fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

749 
text{*A fact for the mutilated chess board*} 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

750 
lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

751 
apply (case_tac "n=0", simp) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

752 
apply (induct "m" rule: nat_less_induct) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

753 
apply (case_tac "Suc (na) <n") 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

754 
(* case Suc(na) < n *) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

755 
apply (frule lessI [THEN less_trans], simp add: less_not_refl3) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

756 
(* case n \<le> Suc(na) *) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

757 
apply (simp add: linorder_not_less le_Suc_eq mod_geq) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

758 
apply (auto simp add: Suc_diff_le le_mod_geq) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

759 
done 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

760 

fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

761 

25942  762 
subsubsection{*The Divides Relation*} 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

763 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

764 
lemma dvdI [intro?]: "n = m * k ==> m dvd n" 
22718  765 
unfolding dvd_def by blast 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

766 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

767 
lemma dvdE [elim?]: "!!P. [m dvd n; !!k. n = m*k ==> P] ==> P" 
22718  768 
unfolding dvd_def by blast 
13152  769 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

770 
lemma dvd_0_right [iff]: "m dvd (0::nat)" 
22718  771 
unfolding dvd_def by (blast intro: mult_0_right [symmetric]) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

772 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

773 
lemma dvd_0_left: "0 dvd m ==> m = (0::nat)" 
22718  774 
by (force simp add: dvd_def) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

775 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

776 
lemma dvd_0_left_iff [iff]: "(0 dvd (m::nat)) = (m = 0)" 
22718  777 
by (blast intro: dvd_0_left) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

778 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24268
diff
changeset

779 
declare dvd_0_left_iff [noatp] 
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24268
diff
changeset

780 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

781 
lemma dvd_1_left [iff]: "Suc 0 dvd k" 
22718  782 
unfolding dvd_def by simp 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

783 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

784 
lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)" 
22718  785 
by (simp add: dvd_def) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

786 

25942  787 
lemmas dvd_refl [simp] = semiring_div_class.dvd_refl [of "m\<Colon>nat", standard] 
788 
lemmas dvd_trans [trans] = semiring_div_class.dvd_trans [of "m\<Colon>nat" n p, standard] 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

789 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

790 
lemma dvd_anti_sym: "[ m dvd n; n dvd m ] ==> m = (n::nat)" 
22718  791 
unfolding dvd_def 
792 
by (force dest: mult_eq_self_implies_10 simp add: mult_assoc mult_eq_1_iff) 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

793 

23684
8c508c4dc53b
introduced (auxiliary) class dvd_mod for more convenient code generation
haftmann
parents:
23162
diff
changeset

794 
text {* @{term "op dvd"} is a partial order *} 
8c508c4dc53b
introduced (auxiliary) class dvd_mod for more convenient code generation
haftmann
parents:
23162
diff
changeset

795 

25942  796 
interpretation dvd: order ["op dvd" "\<lambda>n m \<Colon> nat. n dvd m \<and> n \<noteq> m"] 
23684
8c508c4dc53b
introduced (auxiliary) class dvd_mod for more convenient code generation
haftmann
parents:
23162
diff
changeset

797 
by unfold_locales (auto intro: dvd_trans dvd_anti_sym) 
8c508c4dc53b
introduced (auxiliary) class dvd_mod for more convenient code generation
haftmann
parents:
23162
diff
changeset

798 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

799 
lemma dvd_add: "[ k dvd m; k dvd n ] ==> k dvd (m+n :: nat)" 
22718  800 
unfolding dvd_def 
801 
by (blast intro: add_mult_distrib2 [symmetric]) 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

802 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

803 
lemma dvd_diff: "[ k dvd m; k dvd n ] ==> k dvd (mn :: nat)" 
22718  804 
unfolding dvd_def 
805 
by (blast intro: diff_mult_distrib2 [symmetric]) 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

806 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

807 
lemma dvd_diffD: "[ k dvd mn; k dvd n; n\<le>m ] ==> k dvd (m::nat)" 
22718  808 
apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst]) 
809 
apply (blast intro: dvd_add) 

810 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

811 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

812 
lemma dvd_diffD1: "[ k dvd mn; k dvd m; n\<le>m ] ==> k dvd (n::nat)" 
22718  813 
by (drule_tac m = m in dvd_diff, auto) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

814 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

815 
lemma dvd_mult: "k dvd n ==> k dvd (m*n :: nat)" 
22718  816 
unfolding dvd_def by (blast intro: mult_left_commute) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

817 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

818 
lemma dvd_mult2: "k dvd m ==> k dvd (m*n :: nat)" 
22718  819 
apply (subst mult_commute) 
820 
apply (erule dvd_mult) 

821 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

822 

17084
fb0a80aef0be
classical rules must have names for ATP integration
paulson
parents:
16796
diff
changeset

823 
lemma dvd_triv_right [iff]: "k dvd (m*k :: nat)" 
22718  824 
by (rule dvd_refl [THEN dvd_mult]) 
17084
fb0a80aef0be
classical rules must have names for ATP integration
paulson
parents:
16796
diff
changeset

825 

fb0a80aef0be
classical rules must have names for ATP integration
paulson
parents:
16796
diff
changeset

826 
lemma dvd_triv_left [iff]: "k dvd (k*m :: nat)" 
22718  827 
by (rule dvd_refl [THEN dvd_mult2]) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

828 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

829 
lemma dvd_reduce: "(k dvd n + k) = (k dvd (n::nat))" 
22718  830 
apply (rule iffI) 
831 
apply (erule_tac [2] dvd_add) 

832 
apply (rule_tac [2] dvd_refl) 

833 
apply (subgoal_tac "n = (n+k) k") 

834 
prefer 2 apply simp 

835 
apply (erule ssubst) 

836 
apply (erule dvd_diff) 

837 
apply (rule dvd_refl) 

838 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

839 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

840 
lemma dvd_mod: "!!n::nat. [ f dvd m; f dvd n ] ==> f dvd m mod n" 
22718  841 
unfolding dvd_def 
842 
apply (case_tac "n = 0", auto) 

843 
apply (blast intro: mod_mult_distrib2 [symmetric]) 

844 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

845 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

846 
lemma dvd_mod_imp_dvd: "[ (k::nat) dvd m mod n; k dvd n ] ==> k dvd m" 
22718  847 
apply (subgoal_tac "k dvd (m div n) *n + m mod n") 
848 
apply (simp add: mod_div_equality) 

849 
apply (simp only: dvd_add dvd_mult) 

850 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

851 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

852 
lemma dvd_mod_iff: "k dvd n ==> ((k::nat) dvd m mod n) = (k dvd m)" 
22718  853 
by (blast intro: dvd_mod_imp_dvd dvd_mod) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

854 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

855 
lemma dvd_mult_cancel: "!!k::nat. [ k*m dvd k*n; 0<k ] ==> m dvd n" 
22718  856 
unfolding dvd_def 
857 
apply (erule exE) 

858 
apply (simp add: mult_ac) 

859 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

860 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

861 
lemma dvd_mult_cancel1: "0<m ==> (m*n dvd m) = (n = (1::nat))" 
22718  862 
apply auto 
863 
apply (subgoal_tac "m*n dvd m*1") 

864 
apply (drule dvd_mult_cancel, auto) 

865 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

866 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

867 
lemma dvd_mult_cancel2: "0<m ==> (n*m dvd m) = (n = (1::nat))" 
22718  868 
apply (subst mult_commute) 
869 
apply (erule dvd_mult_cancel1) 

870 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

871 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

872 
lemma mult_dvd_mono: "[ i dvd m; j dvd n] ==> i*j dvd (m*n :: nat)" 
22718  873 
apply (unfold dvd_def, clarify) 
874 
apply (rule_tac x = "k*ka" in exI) 

875 
apply (simp add: mult_ac) 

876 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

877 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

878 
lemma dvd_mult_left: "(i*j :: nat) dvd k ==> i dvd k" 
22718  879 
by (simp add: dvd_def mult_assoc, blast) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

880 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

881 
lemma dvd_mult_right: "(i*j :: nat) dvd k ==> j dvd k" 
22718  882 
apply (unfold dvd_def, clarify) 
883 
apply (rule_tac x = "i*k" in exI) 

884 
apply (simp add: mult_ac) 

885 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

886 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

887 
lemma dvd_imp_le: "[ k dvd n; 0 < n ] ==> k \<le> (n::nat)" 
22718  888 
apply (unfold dvd_def, clarify) 
889 
apply (simp_all (no_asm_use) add: zero_less_mult_iff) 

890 
apply (erule conjE) 

891 
apply (rule le_trans) 

892 
apply (rule_tac [2] le_refl [THEN mult_le_mono]) 

893 
apply (erule_tac [2] Suc_leI, simp) 

894 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

895 

25942  896 
lemmas dvd_eq_mod_eq_0 = dvd_def_mod [of "k\<Colon>nat" n, standard] 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

897 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

898 
lemma dvd_mult_div_cancel: "n dvd m ==> n * (m div n) = (m::nat)" 
22718  899 
apply (subgoal_tac "m mod n = 0") 
900 
apply (simp add: mult_div_cancel) 

901 
apply (simp only: dvd_eq_mod_eq_0) 

902 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

903 

21408  904 
lemma le_imp_power_dvd: "!!i::nat. m \<le> n ==> i^m dvd i^n" 
22718  905 
apply (unfold dvd_def) 
906 
apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst]) 

907 
apply (simp add: power_add) 

908 
done 

21408  909 

26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

910 
lemma mod_add_left_eq: "((a::nat) + b) mod c = (a mod c + b) mod c" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

911 
apply (rule trans [symmetric]) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

912 
apply (rule mod_add1_eq, simp) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

913 
apply (rule mod_add1_eq [symmetric]) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

914 
done 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

915 

fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

916 
lemma mod_add_right_eq: "(a+b) mod (c::nat) = (a + (b mod c)) mod c" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

917 
apply (rule trans [symmetric]) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

918 
apply (rule mod_add1_eq, simp) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

919 
apply (rule mod_add1_eq [symmetric]) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

920 
done 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

921 

25162  922 
lemma nat_zero_less_power_iff [simp]: "(x^n > 0) = (x > (0::nat)  n=0)" 
22718  923 
by (induct n) auto 
21408  924 

925 
lemma power_le_dvd [rule_format]: "k^j dvd n > i\<le>j > k^i dvd (n::nat)" 

22718  926 
apply (induct j) 
927 
apply (simp_all add: le_Suc_eq) 

928 
apply (blast dest!: dvd_mult_right) 

929 
done 

21408  930 

931 
lemma power_dvd_imp_le: "[i^m dvd i^n; (1::nat) < i] ==> m \<le> n" 

22718  932 
apply (rule power_le_imp_le_exp, assumption) 
933 
apply (erule dvd_imp_le, simp) 

934 
done 

21408  935 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

936 
lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)" 
22718  937 
by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def) 
17084
fb0a80aef0be
classical rules must have names for ATP integration
paulson
parents:
16796
diff
changeset

938 

22718  939 
lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1] 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

940 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

941 
(*Loses information, namely we also have r<d provided d is nonzero*) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

942 
lemma mod_eqD: "(m mod d = r) ==> \<exists>q::nat. m = r + q*d" 
22718  943 
apply (cut_tac m = m in mod_div_equality) 
944 
apply (simp only: add_ac) 

945 
apply (blast intro: sym) 

946 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

947 

13152  948 
lemma split_div: 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

949 
"P(n div k :: nat) = 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

950 
((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

951 
(is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))") 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

952 
proof 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

953 
assume P: ?P 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

954 
show ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

955 
proof (cases) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

956 
assume "k = 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

957 
with P show ?Q by(simp add:DIVISION_BY_ZERO_DIV) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

958 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

959 
assume not0: "k \<noteq> 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

960 
thus ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

961 
proof (simp, intro allI impI) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

962 
fix i j 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

963 
assume n: "n = k*i + j" and j: "j < k" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

964 
show "P i" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

965 
proof (cases) 
22718  966 
assume "i = 0" 
967 
with n j P show "P i" by simp 

13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

968 
next 
22718  969 
assume "i \<noteq> 0" 
970 
with not0 n j P show "P i" by(simp add:add_ac) 

13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

971 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

972 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

973 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

974 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

975 
assume Q: ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

976 
show ?P 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

977 
proof (cases) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

978 
assume "k = 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

979 
with Q show ?P by(simp add:DIVISION_BY_ZERO_DIV) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

980 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

981 
assume not0: "k \<noteq> 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

982 
with Q have R: ?R by simp 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

983 
from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"] 
13517  984 
show ?P by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

985 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

986 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

987 

13882  988 
lemma split_div_lemma: 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

989 
assumes "0 < n" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

990 
shows "n * q \<le> m \<and> m < n * Suc q \<longleftrightarrow> q = ((m\<Colon>nat) div n)" (is "?lhs \<longleftrightarrow> ?rhs") 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

991 
proof 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

992 
assume ?rhs 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

993 
with mult_div_cancel have nq: "n * q = m  (m mod n)" by simp 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

994 
then have A: "n * q \<le> m" by simp 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

995 
have "n  (m mod n) > 0" using mod_less_divisor assms by auto 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

996 
then have "m < m + (n  (m mod n))" by simp 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

997 
then have "m < n + (m  (m mod n))" by simp 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

998 
with nq have "m < n + n * q" by simp 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

999 
then have B: "m < n * Suc q" by simp 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

1000 
from A B show ?lhs .. 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

1001 
next 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

1002 
assume P: ?lhs 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

1003 
then have "divmod_rel m n q (m  n * q)" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

1004 
unfolding divmod_rel_def by (auto simp add: mult_ac) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

1005 
then show ?rhs using divmod_rel by (rule divmod_rel_unique_div) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

1006 
qed 
13882  1007 

1008 
theorem split_div': 

1009 
"P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or> 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1010 
(\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))" 
13882  1011 
apply (case_tac "0 < n") 
1012 
apply (simp only: add: split_div_lemma) 

1013 
apply (simp_all add: DIVISION_BY_ZERO_DIV) 

1014 
done 

1015 

13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1016 
lemma split_mod: 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1017 
"P(n mod k :: nat) = 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1018 
((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1019 
(is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))") 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1020 
proof 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1021 
assume P: ?P 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1022 
show ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1023 
proof (cases) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1024 
assume "k = 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1025 
with P show ?Q by(simp add:DIVISION_BY_ZERO_MOD) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1026 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1027 
assume not0: "k \<noteq> 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1028 
thus ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1029 
proof (simp, intro allI impI) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1030 
fix i j 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1031 
assume "n = k*i + j" "j < k" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1032 
thus "P j" using not0 P by(simp add:add_ac mult_ac) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1033 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1034 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1035 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1036 
assume Q: ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1037 
show ?P 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1038 
proof (cases) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1039 
assume "k = 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1040 
with Q show ?P by(simp add:DIVISION_BY_ZERO_MOD) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1041 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1042 
assume not0: "k \<noteq> 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1043 
with Q have R: ?R by simp 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1044 
from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"] 
13517  1045 
show ?P by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1046 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1047 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1048 

13882  1049 
theorem mod_div_equality': "(m::nat) mod n = m  (m div n) * n" 
1050 
apply (rule_tac P="%x. m mod n = x  (m div n) * n" in 

1051 
subst [OF mod_div_equality [of _ n]]) 

1052 
apply arith 

1053 
done 

1054 

22800  1055 
lemma div_mod_equality': 
1056 
fixes m n :: nat 

1057 
shows "m div n * n = m  m mod n" 

1058 
proof  

1059 
have "m mod n \<le> m mod n" .. 

1060 
from div_mod_equality have 

1061 
"m div n * n + m mod n  m mod n = m  m mod n" by simp 

1062 
with diff_add_assoc [OF `m mod n \<le> m mod n`, of "m div n * n"] have 

1063 
"m div n * n + (m mod n  m mod n) = m  m mod n" 

1064 
by simp 

1065 
then show ?thesis by simp 

1066 
qed 

1067 

1068 

25942  1069 
subsubsection {*An ``induction'' law for modulus arithmetic.*} 
14640  1070 

1071 
lemma mod_induct_0: 

1072 
assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)" 

1073 
and base: "P i" and i: "i<p" 

1074 
shows "P 0" 

1075 
proof (rule ccontr) 

1076 
assume contra: "\<not>(P 0)" 

1077 
from i have p: "0<p" by simp 

1078 
have "\<forall>k. 0<k \<longrightarrow> \<not> P (pk)" (is "\<forall>k. ?A k") 

1079 
proof 

1080 
fix k 

1081 
show "?A k" 

1082 
proof (induct k) 

1083 
show "?A 0" by simp  "by contradiction" 

1084 
next 

1085 
fix n 

1086 
assume ih: "?A n" 

1087 
show "?A (Suc n)" 

1088 
proof (clarsimp) 

22718  1089 
assume y: "P (p  Suc n)" 
1090 
have n: "Suc n < p" 

1091 
proof (rule ccontr) 

1092 
assume "\<not>(Suc n < p)" 

1093 
hence "p  Suc n = 0" 

1094 
by simp 

1095 
with y contra show "False" 

1096 
by simp 

1097 
qed 

1098 
hence n2: "Suc (p  Suc n) = pn" by arith 

1099 
from p have "p  Suc n < p" by arith 

1100 
with y step have z: "P ((Suc (p  Suc n)) mod p)" 

1101 
by blast 

1102 
show "False" 

1103 
proof (cases "n=0") 

1104 
case True 

1105 
with z n2 contra show ?thesis by simp 

1106 
next 

1107 
case False 

1108 
with p have "pn < p" by arith 

1109 
with z n2 False ih show ?thesis by simp 

1110 
qed 

14640  1111 
qed 
1112 
qed 

1113 
qed 

1114 
moreover 

1115 
from i obtain k where "0<k \<and> i+k=p" 

1116 
by (blast dest: less_imp_add_positive) 

1117 
hence "0<k \<and> i=pk" by auto 

1118 
moreover 

1119 
note base 

1120 
ultimately 

1121 
show "False" by blast 

1122 
qed 

1123 

1124 
lemma mod_induct: 

1125 
assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)" 

1126 
and base: "P i" and i: "i<p" and j: "j<p" 

1127 
shows "P j" 

1128 
proof  

1129 
have "\<forall>j<p. P j" 

1130 
proof 

1131 
fix j 

1132 
show "j<p \<longrightarrow> P j" (is "?A j") 

1133 
proof (induct j) 

1134 
from step base i show "?A 0" 

22718  1135 
by (auto elim: mod_induct_0) 
14640  1136 
next 
1137 
fix k 

1138 
assume ih: "?A k" 

1139 
show "?A (Suc k)" 

1140 
proof 

22718  1141 
assume suc: "Suc k < p" 
1142 
hence k: "k<p" by simp 

1143 
with ih have "P k" .. 

1144 
with step k have "P (Suc k mod p)" 

1145 
by blast 

1146 
moreover 

1147 
from suc have "Suc k mod p = Suc k" 

1148 
by simp 

1149 
ultimately 

1150 
show "P (Suc k)" by simp 

14640  1151 
qed 
1152 
qed 

1153 
qed 

1154 
with j show ?thesis by blast 

1155 
qed 

1156 

3366  1157 
end 